First Principles Predictions of Stacking Fault Properties in high-Mn Steels

High-Mn-steels are excellent candidates for the next generation of light-weight high-strength steels due to their exceptional mechanical characteristics. The mechanical properties of such steels sensitively depend on their microstructure. The stacking faults that occur in the austenitic phase of high-Mn steels are of particular importance since their energies define which plasticity mechanism (twinning induced or transformation induced plasticity, TWIP or TRIP) prevail in the steel [1]. Up to now the theoretical understanding of such defects was based on a combination of the regular and subregular solution models and the Olson-Cohen approach [2]. The parameters entering these models are either derived from explicit measurements of the stacking fault energies, or from empirical relations for the Gibbs free energy G of fcc and hcp phases of the material [3]. Such an approach relies on the continuum description of individual fcc and hcp phases and does not account for the atomistic structure of the stacking faults. However, in order to effectively explore chemical trends, to deliver parameters for phenomenological models, and to identify new routes to optimize high-Mn-steels an atomistic understanding of such structural defects is decisive. The ultimate goal of the current research is the development of a generalized first principle theoretical framework that allows an accurate temperature- and composition-dependent description of the stacking fault properties in various types of steels, and the application of the method to the particular case of high-Mn steels. The framework is based on density functional theory (DFT), which is based on the explicit description of the chemical bonding between atoms and is an established tool for the deriving material properties. Since Fe and Mn chemical species are the two dominating types of atoms in realistic high-Mn-steels, we first start with the description of the stacking faults in the FeMn-alloy. The actual steels can be obtained by admixing small concentration of other chemical elements like, e.g., Carbon, Si, etc. substitutionally or interstitially in this host structure. Austenitic FeMn-alloys are ordered antiferromagnetically (aFM), with the Néel temperature being ~350-400 K at Mn contents larger than 20 atomic percents [5]. Information regarding the actual atomic and magnetic structure, i.e. whether there exist short-range or long-range order, is lacking. To calculate the stacking fault energies we, therefore, first had to identify the actual atomic arrangement as function of temperature and chemical composition. The existing first principle calculations of the SFE can be divided into two classes: They are either performed for materials at their ground state at T=0 K (see, e.g. Ref. [6]-[8]), i.e. completely neglecting the temperature dependence of the SFE, or temperature-dependent calculations for the paramagnetic disordered phase [9]. The approach employed in the latter calculations is, however, not applicable to steels where atomic relaxations are significant. For example, it cannot account for changes in the structure that may occur at the stacking faults, and is limited to high temperatures where no ordering exists. In order to calculate the SFE in FeMn-alloys over a wide range of temperatures and compositions it is necessary, therefore, to develop a new methodology that is able to include (i) the change of the atomic (magnetic) structure as function of temperature and composition, (ii) provides an effective method for evaluating the SFE for a given representative structure, and (iii) is able to accurately account for the temperature effects. Below we present the first steps towards constructing such an approach on the example of FeMn-alloys. In order to estimate the importance of the magnetic interactions we first switched them off and treated the alloy as nonmagnetic. The only structural unknown is then the chemical structure. To identify it over the complete range of Mn compositions we considered two distinct extreme cases: (i) the low temperature limit, where chemically ordered structures are energetically most stable, and (ii) the high temperature limit, where entropy contributions are large and completely destroy any ordering in the system, resulting in the chemically disordered alloy.

The ground-state atomic configurations corresponding to the low temperature limit have been indentified by the cluster-expansion method [10], which allows an effective mapping of the first principles DFT calculations onto a model Hamiltonian. The model Hamiltonian has then been used for calculating the total energies of numerous crystal structures with different arrangements of Fe and Mn atomic species. Employing this method we have considered more than 5000 distinct atomic configurations without loss of accuracy, and reliably identified the true ground state structures (A1, L13, L10) of FeMn-alloy over the entire range of Mn concentrations (see Fig. 1). In a second step we used these ground state structures to simulate the so-called generalized stacking fault energy (γ-surface). It is obtained by shifting two half crystals of the austenitic material with respect to each other along the (111) glide plane by a given displacement vector, and calculating the SFE Γ as an excess energy that is conserved in the defect area:

Γ=(Gdef-Gideal)/Aint. (1)

Here, Gdef is the free energy of a crystal containing the defect, Gideal is that of the ideal fcc crystal, and Aint is the area over which the stacking fault extends in the (111) plane. The positions of the global minima on the calculated γ-surface correspond to intrinsic stacking faults which realize the fcc/hcp transition in the crystal due to slips. A comparison with available empirical data (see Fig. 2) indicates that the calculated SFE for nonmagnetic ordered alloys does not reproduce the available empirical data: while the empirical SFE decreases with increasing Mn content at low Mn concentrations, goes through a minimum and then increases with higher Mn-concentrations, the calculated SFE monotonously increases with increasing Mn content. Further, the predicted SFE are largely negative (below ~ -300 mJ/m2) while the extrapolation of the empirical data to T=0 K results in values of the SFE not lower than -40 mJ/m2. This large discrepancy indicates that magnetic interactions and/or chemical ordering play a crucial role in FeMn-alloys and have to be taken into account in the simulations.

Fig. 2: (a) Empirical data for SFE austenitic FeMn-alloys as a function of Mn content at room temperature (b) Summary of the corresponding calculated intrinsic SFE at T=0 K. The high temperature magnetic phase (paramagnetic chemically disordered alloys), as well as the nonmagnetic phase at low temperature (chemically ordered alloys) and at high temperature limit (chemically disordered alloys) are shown. The results for the chemically ordered phase are obtained by explicitly calculating the γ-surface. For the various disordered phases - the 1st-order ANNNI model (see text) has been employed.

In order to identify which of the two remaining mechanisms is responsible for the observed discrepancy we calculated in a next step instrinsic SFEs for the disordered (high temperature) phase of nonmagnetic FeMn-alloys. To represent random alloys we utilized the concept of special quasirandom structures (SQS) [11]. This method guarantees the best possible description of an ideal random structure for a given number of atoms in a periodically repeated supercell, and allows to properly include atomic relaxations. To calculate the SFE for random alloys within this approach, we had to reduce the computational complexity and employed the axial next-nearest-neighbor Ising (ANNNI) model [12]. This model had already been successfully applied to austenitic stainless steels [9]. The ANNNI model describes the actual intrinsic stacking fault from a series expansion of different infinitely repeated stacking sequences. In the first order (Eq. (2)) only the energies of the fcc and hcp stacking sequences are considered, while higher-order expansions incorporate more sophisticated phases, like, e.g. the double hcp phase (Eq. (3)):

Γ1=2(Ghcp-Gfcc)/Aint,, (2)

Γ2=(Ghcp+2Gdhcp-3Gfcc)/Aint. (3)

The results for the ISFE calculated for disordered nonmagnetic FeMn-alloys are shown in Fig. 2(b). In contrast to the low temperature case where only ordered structures have been assumed, the qualitative agreement with the empirical data is improved: The SFE no longer monotonously increase with the Mn concentration, but correctly reproduce the minimum of the SFE at medium Mn content (cMn≈25%). Quantitatively, however, the agreement is less satisfactory and is similar to a case of the ordered nonmagnetic alloy.

Because the type of the antiferromagnetic ordering (e.g., layer-wise aFM, double layer aFM, spin-spiral state), as well as the chemical order-disorder transition temperature is not known from experiment, we consider first the paramagnetic phase of the FeMn-alloys and assume that no chemical order exists in the system. Such a paramagnetic chemically disordered alloy corresponds then to the high temperature state of the crystal. To simulate such a scenario we have employed a similar approach as used for nonmagnetic disordered alloys, i.e. a combination of the SQS and the ANNNI model: the paramagnetic state of the system was simulated with quasiquaternary alloys Fe↑x/2Fe↓x/2Mn↑(1-x)/2Mn↓(1-x)/2, where a collinear magnetic structure is assumed and ↑,↓ correspond to up and down spin channels of the system. This approach is formally equivalent to the disordered local moment (DLM) model, that is known to provide a reliable description of the paramagnetic state of the crystal [9]. The results of the corresponding calculations are shown in Fig. 2(b) by the blue dots. They clearly indicate that magnetic interactions indeed play an essential role in FeMn-alloys and cannot be disregarded. Specifically, the ISFE are increased for all compositions when compared with the corresponding nonmagnetic structures, with the most pronounced magnitude of the increase at low Mn-content. Further, the experimentally observed presence of the SFE minimum at Mn-concentrations less than 50% is reproduced. Still, although the quantitative agreement between the first principles and empirical data has been significantly improved by accurately taking into account magnetism and chemical disorder, the present level of accuracy is still not sufficient for qualitative prediction of the plastic properties of FeMn-alloys, for which errors in the order of several mJ/m2 may become important.

To identify the source of the remaining discrepancy we restrict here on the case of pure fcc Fe, where no complication due to unknown chemical order exist. We note first that the calculated SFE of the paramagnetic disordered phase corresponds to T=0 K. In reality, however, this state is realized only at elevated temperatures. For an adequate comparison, therefore, it is necessary to account for the temperature dependence of the SFE. In the framework of the ANNNI model the problem substantially simplifies and requires only a description of the temperature-dependent free energy G(T) of bulk crystals with different stacking sequences. Although it is principally possible to calculate G(T) explicitly (see, e.g., highlight #Highlight_TILMANN-HICKEL,FRITZ-KOERMANN,BLAZEJ-GRABOWSKI#), we employ here a more approximate treatment of the temperature effects. Specifically, we explicitly include the volume change temperature by extracting the thermal linear expansion coefficient from experimental data and extrapolating it to low temperature [13]. We further include the effects of magnetic entropy via the mean-field expression:

Smag(V)=Σ kB ln(2Si(V)+1), (4)

where Si(V) is the volume dependent spin quantum number of the atom i, which can be related to the corresponding localized atomic magnetic moment mi(V)=2Si(V) [9]. The contribution due to vibrational entropy is not included. The obtained estimate of the free energy G(T)=E(V(T))-TSmag(V(T)) corresponds to the upper limit for both the magnetic entropy and the thermal volume expansion.

Fig. 3: Temperature dependence of the intrinsic stacking fault energy in paramagnetic fcc Fe obtained when the thermal volume expansion and the contribution of the magnetic excitations are included. The empirical dependence is taken from Ref. 3.

In Fig. 3 we show the comparison of the temperature-dependent theoretical calculations for the SFE in paramagnetic fcc Fe with an empirical relation for the SFE in FeMn-alloys at the limiting case of zero manganese content. The calculated SFE sensitively depend on the temperature, and, in agreement with the empirical curve, correctly predicts a nonlinear increase of the SFE with temperature. However, because the magnitude of the SFE at T=0 K in our calculations is ~130 mJ/m2 smaller than the corresponding empirical value, the temperature effects are not strong enough to fully reproduce the experiment. Our calculations further show (Fig. 3), that although the magnetic excitations are significant, the thermal volume expansion dominates the increase of the SFE with increasing temperature. Such pronounced volume dependence is unexpected and was assumed in previous studies to be negligible [9]. We note, that in actual alloys the volume might change not only due to an increase of the temperature, but might occur due to adding other species like e.g. Carbon, applying external strain/stress, or in the vicinity of local strain fields.

In summary, we have presented a generalized first principle approach for calculation of the stacking fault energy in realistic steels. The method is based on a multiphysics combination of approaches from alloy physics, microstructure physics, physics of magnetism, and thermodynamics. We have briefly highlighted the method for calculating the SFE for high- and low-temperature phases of nonmagnetic binary alloys, as well as a method for calculating the high-temperature phase of the magnetic binary alloys. For the considered case of FeMn-alloys our results showed a hitherto unexpected conclusion: In contrast to conventional wisdom that stacking fault related plasticity mechanisms can be described by a single value for the SFE, the results clearly show that the volume-dependence of the stacking fault energy should be included. An important consequence is that the prevailing plasticity mechanisms (TRIP/TWIP) can even change upon deformation of FeMn-alloys. These results indicate that a careful revision of the available experimental data should be made. For this a set of reference experiments on SFE in single-crystal FeMn-alloy will be performed in collaboration with the MU department (Prof. Raabe). We finally note that the presented approach is not limited to binary alloys, and allows a straightforward generalization to multicomponent systems, including lattice mismatch alloys where atomic relaxations become essential. Further development of the model towards identification of the stacking fault properties at the intermediate temperature range where chemical and/or magnetic ordering exist, as well as a more accurate inclusion of the various temperature contributions to the SFE, is still required and is subject of ongoing research.